ARTICLE IN PRESS
Physica B 374–375 (2006) 423–425
www.elsevier.com/locate/physb
Muonium diffusion dynamics in mercury oxide
J. Piroto Duartea,, J.M. Gila, H.V. Albertoa, R.C. Vilãoa, A. Weidingera,
N. Ayres de Camposa, S.F.J. Coxb,c, J.S. Lordc, S.P. Cottrellc, E.A. Davisd
a
Physics Department, University of Coimbra, P-3004-516 Coimbra, Portugal
b
Physics Department, University College London, WC1E 6BT, UK
c
ISIS Facility, Rutherford Appleton Laboratory, Chilton, Didcot, Oxon OX11 0QX, UK
d
Department of Materials Science and Metallurgy, University of Cambridge, CB2 3QZ, UK
Abstract
The diffusion dynamics of the neutral muonium state found in HgO is addressed in this work. We propose a hopping model for the
diffusion, and use it to analyse time-domain mSR data. It is found that the diffusion is an incoherent quantum process, with an activation
energy of 5.2(2) meV. The analysis also points to the anti-bonding site as the best-suited candidate for the muon’s localisation.
r 2005 Elsevier B.V. All rights reserved.
Keywords: II–VI Semiconductors; Mercury oxide; Muonium diffusion
1. Introduction
The understanding of the role of hydrogen in semiconductors has developed considerably in recent years since it
was discovered that interstitial hydrogen could act as a
shallow-donor impurity in compound semiconductors
[1–4]. The mSR technique played a crucial role in this
discovery; indeed, the very first experimental evidence was
obtained with mSR experiments on cadmium sulphide and
zinc oxide, where an unusually low hyperfine interaction
associated with the muonium (Mu) state was identified
[1,2]. Since then, systematic surveys of other compound
semiconductors, mostly oxides, have enlarged the list of
host systems in which the shallow-level donor muonium
state is formed [5,6].
Within the context of a deep-level/shallow-level dual
picture for hydrogen impurity states in semiconductors, the
neutral muonium state found below 150 K in the wide-gap
semiconductor mercury oxide appears somewhat as a
further novelty [7,8]. Its hyperfine coupling parameters
Aiso ¼ 14.93 MHz and D ¼ 5.2 MHz place it mid-way
between the bond-centre muonium deep states known for
long in the elemental and III–V compound semiconductors
Corresponding author. Tel.: +351 239 410 685; fax: +351 239 829 158.
E-mail address: [email protected] (J. Piroto Duarte).
0921-4526/$ - see front matter r 2005 Elsevier B.V. All rights reserved.
doi:10.1016/j.physb.2005.11.122
(Aiso and D several tens of MHz), and the anion antibonding shallow states recently discovered (Aiso and D
hundreds of kHz). The uniqueness of this state arises
from the unusual structure of HgO, shaped by broken
chains running parallel to the a-axis that coordinate the Hg
and O atoms (see Fig. 1), contrasting with the much more
regular tetrahedral coordination of most semiconductor
systems. This makes controversial whether the neutral
muonium state formed in HgO is located at the bondcentre or at the oxygen’s anti-bonding site, and no
experimental evidence clearing this matter has been put
forward so far.
This neutral muonium state in HgO has also been found
to undergo diffusion through crystallographically equivalent positions below ionisation. The existence of diffusion
dynamics has been inferred from data taken with powder
samples which imply averaging of the state’s anisotropy [7].
In high-field TF geometry measurements, increasing
temperature produces a gradual narrowing of the frequency powder-pattern lineshape characteristic of axially
symmetric muonium; in LF repolarisation measurements,
the DM ¼ 1 level-crossing resonance dip narrows from an
asymmetric powder-shape at low temperatures to a more
symmetrical resonance at higher ones.
This work reports a quantitative mSR study performed
on the diffusion dynamics of muonium in HgO. Existing
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Fig. 1. The (i) bond-centre and (ii) anti-bonding diffusion paths
considered in our model for muonium diffusion in HgO. Hg atoms are
represented by small, darker spheres, O atoms by large, brighter spheres
and Mu by small white spheres.
and new experimental data taken on powder samples with
the GPS (TF geometry) and EMU (LF geometry) spectrometers at PSI and ISIS respectively are analysed with a
simplified hopping model, and its results are discussed.
position in the paths; the real hop rate will be the double
of L.
The time evolution of the muon’s polarisation in LF and
TF geometries was computed using the equations of
motion for all polarisations of two muonium systems with
different hyperfine interaction tensors undergoing a reversible transition at a rate L [9]. The procedure followed is a
generalisation of the Wangsness–Bloch equations for a
single muonium system to the two system case, adopting
the remarks described in Ref. [10]. The two hyperfine
tensors were constrained to have the same hyperfine
parameters Aiso ¼ 14.93 MHz and D ¼ 5.2 MHz, and
symmetry axis concurrent at 107.31. Powder averaging
was performed numerically with the (also numerical)
solutions of the two-muonium system equations. A
phenomenological electronic depolarisation rate n was
included as well, in order to model the nuclear hyperfine
interaction with the surrounding spin-carrying Hg nuclei
(17% S ¼ 12, 13% S ¼ 32).
Simulations performed with this model were found to
reproduce the main features observed in the diffusion data
above 10 K, namely the expected onset of the powderdistribution narrowing in high TF and of the narrowing of
the LF resonance dip to a symmetric shape at a jump rate
near the value of the anisotropy parameter. This study also
showed that the high-TF and LF repolarisation data
should exhibit a very low sensitivity to the hop rate L
outside that region. It was found as well that the highest
sensitivity to L is achieved with data collected in the socalled Mu* LF magic field [9,10], at which the amount of
information regarding L in the time spectra is maximal
thanks to the non-existence of damping of the f12
precession by the powder distribution.
3. Results and discussion
2. Diffusion model
Presuming the observed muonium state has the same
symmetry axis as the Hg–O bond, hopping paths through
bond-centre or anti-bonding sites in adjacent chains
provide the anisotropy averaging observed in Ref. [7]
(Fig. 1). Careful examination shows that in each path
jumps take place between positions either with the same
symmetry axis, or with symmetry axis concurrent at 107.31,
the Hg–O–Hg bond angle. For the sake of simplicity, one
may start by assuming that all polarisations of the
muonium system (muon and electron) do not suffer any
change when it jumps between sites having the same
symmetry axis. This reduces the hopping problem to the
reversible transition between two muonium states with the
same hyperfine parameters, but different symmetry axis, at
a hopping rate L. We note that these considerations apply
to both the bond-centre and the oxygen anti-bonding
paths, and that therefore the model makes intrinsically no
distinction between the two possibilities. We also make
clear that the hopping rate L refers to jumps between
positions with different symmetry axis, i.e., every second
Most of the quantitative analysis with the diffusion
model was performed on magic field time-spectrum data
collected at the EMU spectrometer. Fig. 2 shows an
Arrhenius plot of the temperature dependence extracted in
that way for the Mu hop rate (double of L, see Section 2).
Between 6 and 30 K, it is in good agreement with the simple
small-polaron model for incoherent quantum diffusion of
light interstitials [11,12], bearing an activation energy of
Ea ¼ 5.2(2) meV
and
transition
matrix
element
J ¼ 0.065(3) meV. Such small activation energy confirms
the quantum nature of the diffusion process, discarding
classical ‘‘over-the-barrier’’ hopping. The transition matrix
element J is also within the expected range of values for
insulators and semiconductors at low temperatures [12]. At
30 K, the hop rate peaks; muonium is highly mobile at that
temperature, tunneling about 800 times during its 2.2 ms
lifetime. Above 30 K, the hop rate decreases, in what
appears to be a change of diffusion regime. Trapping at a
defect is not the mechanism responsible for this decrease,
since the average length traveled by muonium at 30 K
considering a one-dimensional random walk is only 60 Å, a
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J. Piroto Duarte et al. / Physica B 374–375 (2006) 423–425
200 50 30 20
centre site hypothesis, even taking into account that only
one third of the Hg nuclei have spin. This is in agreement
with recent theoretical results which find electrostatic stable
positions for the muon at sites anti-bonding to the oxygen
atom [13].
Temperature (K)
10
Mu Hop Rate (MHz)
1000
100
Ea = 5.2(2) meV
J = 0.065(3) meV
10
Acknowledgements
1
0.1
0.01
50
100
150
1000/T (K-1)
425
200
250
Fig. 2. Arrhenius plot of the Mu hop rate for the neutral muonium state
in HgO up to 75 K (beyond this temperature, the effects of diffusion
become masked due to ionisation of the muonium state). Bellow 30 K, an
incoherent quantum diffusion process is followed.
distance too small to allow muonium finding any impurity
if one accounts for the high purity of the samples. Trapping
at a grain boundary is ruled out in a similar way.
Insight about the muon site may also be derived from the
magic-field analysis with the diffusion model. The electronic depolarisation rate n was seen to increase with
temperature from zero to 0.25(10) MHz at 10 K, decreasing
thereof again to zero. Since n is expected to model the
nuclear hyperfine interaction with the Hg nuclei, its peak
value may be taken as a guess for that interaction’s
coupling. A rough estimate of the spin-carrying Hg nuclei
to muon distance assuming a 1s type electronic wave
function for the muonium computed with that coupling
amounts to about 6 Å, a value far too large for the bond-
The assistance of all LMU staff at PSI and mSR
instrument scientists at ISIS is gratefully acknowledged.
This work was performed at the Swiss Muon Source, Paul
Scherrer Institute, Villigen, Switzerland, and partially
supported by the European Commission under the Sixth
Framework Programme through the Key Action: Strengthening the European Research Area, Research Infrastructures, contract no: RII3-CT-2004-505925, and the
European Union Framework V access to Research
Infrastructures Programme. Funding has also been provided by the EPSRC grants GMM0041 and GR/R25361
(UK) and the Portuguese Foundation for Science and
Technology (FCT) through grant POCTI/35334/FIS/2000.
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